The adiabatic approximation

In large amplitude nuclear deformation the role of level crossings or pseudo-crossings of intrinsic levels is an outstanding problem. A model consisting of a harmonic oscillator as a collective part and a spin whose orientation energy depends on the collective coordinate is studied in order to clarify this problem. Especially the timeindependent stationary state problem is discussed. Various perturbational expansions are presented and conditions for their convergence are given. The adiabatic approximation leading to a renormalization of the collective mass due to the level crossing is valid for vibrations with energies small compared to intrinsic energy differences.     

Ternary fission in the liquid drop model

The fissioning of a nucleus into three equal fragments is considered in the liquid drop model. Saddlepoint energies for prolate and oblate ternary fission are given as a function of the fissiity parameter.     

Non axial vibrations in the adiabatic approximation

The adiabatic time-dependent Hartree-Fock method, together with a scaling assumption is used for the calculations of the mass parameters associated with non-axial quadrupole (γ) collective vibrations, and for the study of the splitting of the giant quadrupole resonance in deformed nuclei.     

An adiabatic model of asymmetric fission

An adiabatic mechanism leading to asymmetric fission is proposed. The rapid change from symmetric to asymmetric fission as a function of the mass-number is simply explained and related to the transition in the liquid drop model from dumb-bell to cylinder-like shapes of the nucleus at the saddle point.     

Examination of the adiabatic approximation in open systems

We examine the notion of the adiabatic approximation in open systems by applying it to closed systems. Our results shows that the notion is equivalent to the standard adiabatic approximation if the systems are initially in eigenstates, and it leads to a more general expression if the systems are in mixed states.     

Ternary fission within statistical approach

The ternary system with a light nucleus between two heavy fragments is assumed to appear from the binary configuration near scission. The formation of a third light nucleus in the binary system is considered. The calculated charge distributions in spontaneous ternary fission of ²⁵²Cf and in induced ternary fission of ⁵⁶Ni are compared with the available experimental data. The neutron multiplicity from the fission fragments is described. The fine structures of the TKE-mass distribution are predicted.     

Ternary fission of No in collinear configuration

We investigate the collinear ternary fission of the ²⁶⁰No isotope. The calculations are performed in the framework of the three cluster model for all possible accompanied light particles of even mass numbers A=4−52$\text{A}=4-52$. The folding nuclear and Coulomb interaction potentials are used, based on the M3Y-Reid nucleon–nucleon force for the nuclear part. The deformation of the involved fragments and their relative orientations with respect to each other inside the fissioning nuclei are considered. Among all possible fragmentation channels, the suggested most probable channels are indicated as the ones showing a peak in the Q-value and a local minimum in the fragmentation potential, with respect to the mass and charge asymmetries. The indicated favored fragmentation channels from the approximate spherical calculations and those obtained after considering the deformations of the produced fragments are discussed in detail. In addition to the preferred heavy fragments of closed shells, favored prolate ones of high deformations appear when the nuclear deformations are taken into account. Among indicated fifty six favored channels, a collinear ternary fission of the ²⁶⁰No isotope is indicated to be most favored through the fragmentation channels of ¹⁵⁰₅₈Ce+₄¹⁰Be+₄₀¹⁰⁰Zr,₆₀¹⁵²Nd+₄¹²Be+₃₈⁹⁶Sr,₅₈¹⁵⁰Ce+₆¹⁴C+₃₈⁹⁶Sr,₅₈¹⁴⁸Ce+₆¹⁶C+₃₈⁹⁶Sr,₅₄¹⁴⁰Xe+₈²²O+₄₀⁹⁸Zr,₄₂¹⁰⁶Mo+₁₈⁴⁸Ar+₄₂¹⁰⁶Mo and ₄₁¹⁰⁴Nb+₂₀⁵²Ca+₄₁¹⁰⁴Nb.     

New approximation of the energy dependences of total cross sections for proton-induced fission of actinide nuclei

The energy dependences of the cross sections for the fission of ²³²Th to ²³⁹Pu nuclei that is induced by protons of energy in excess of 50 MeV are proposed to be approximated by the sum of two functions. Of these, the first decreases exponentially with increasing proton energy. It represents the contribution of two-step fission, which competes with particle evaporation. The second function decreases exponentially as the energy grows to about 200 MeV, whereupon it increases up to an energy of 1 or 2.5 GeV. After that, it again decreases for ²³²Th to ²³³U nuclei or remains nearly unchanged for ²³⁷Np and ²³⁹Pu nuclei. This function is likely to represent the sum of the contributions from three modes of single-stage fission.     

Asymmetric fission of the pre-actinide nuclei

Study of α particles, protons and neutrons emission in reactions induced by¹⁶O on Sn targets. The threshold for direct a emission is found to be v/c?0.04 (v velocity of the projectile at the top of the Coulomb barrier) whereas direct protons and neutrons, if any, would be emitted for beam energies higher than 7.8MeV/n. (v/c>0.092). Atv/c 0.092, 90% of the direct a cross section is shown to be incomplete fusion. On¹¹⁶Sn target, besides the main exit channels α4n and αp4n, we observed 2α4n and α2p4n channels where the direct α is followed by evaporation particles. This implies a lower initial angular momentum than for the main channels.     

Sufficiency criterion for the validity of the adiabatic approximation

We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it, along with additional conditions, is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all N-dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

The adiabatic approximation in problems in nonlinear optics

The frequency dependence in the adiabatic approximation is examined for the third-rank polarizability tensors of a crystal lacking a center of symmetry. These tensors contain electronic, ionic, and mixed terms, the first type being independent of temperature.I am indebted for discussions to L. M. Belyaev, D. N. Zubarev, S. V. Tyablikov, V. V. Nabatov, Yu. V. Pisarevskii, and Yu. V. Shaldin.     

Test of the validity of the adiabatic approximation in scattering theory

Exact solutions for the scattering of a fast projectile x on deuterium are compared with various versions of the fixed scatterers approximation. The content of the approximations for a nucleon and a different projectile is discussed. The calculations show the approximate agreement of angular distributions in forward direction and the amount of disagreement in the backward hemisphere. In the case of nucleon scattering the importance of nucleon exchange, neglected in the FSA, is explicitly demonstrated. As expected, the quality of the tested approximations rapidly improves with diminishing mass ratio M_x/M_N.     

Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

Neutrino-induced fission of neutron-rich nuclei

We calculate neutrino-induced fission cross sections for selected nuclei with Z=84-92. We show that these reactions populate the daughter nucleus at excitation energies where shell effects are significantly washed out, effectively reducing the fission barrier. If the r process occurs in the presence of a strong neutrino fluence, and electron neutrino average energies are sufficiently high, perhaps as a result of matter-enhanced neutrino flavor transformation, then neutrino-induced fission could lead to significant alteration in the r-process flow in slow outflow scenarios.     

Conductance quantization in graphene nanoribbons: adiabatic approximation

A theory of electron states for graphene nanoribbons with a smoothly varying width is developed. It is demonstrated that the standard adiabatic approximation allowing to neglect the mixing of different standing waves is more restrictive for the massless Dirac fermions in graphene than for the conventional electron gas. For the case of zigzag boundary conditions, one can expect a well-pronounced conductance quantization only for highly excited states. This difference is related to the relativistic Zitterbewegung effect in graphene.     

U + U potential in the sudden and adiabatic approximation

A recent approach for calculating both the real and imaginary parts of the ion-ion potential has been modified to produce the ground state properties of separate nuclei. This approach is used to study both the energy and orientation dependence of the optical potential between two U nuclei adopting the sudden and adiabatic approximations.